SIMPLIFIED MASS TRANSFER RELATIONSHIPS FOR DIFFUSION-

CONTROLLED AIR DEHYDRATION OF REGULAR SOLIDS

H. S. Ramaswamy1 and K. V. Lo2

department of Food Science, and Department of Bio-Resource Engineering, University of British Columbia,Vancouver, B.C. V6T1W5

Received 3 April 1982, accepted 15 September 1982

Ramaswamy, H. S. and K. V. Lo. 1983. Simplified mass transfer relationships for diffusion-controlled air dehydration of regular solids. Can. Agric. Eng. 25:143-148.

Simplified relationships were developed forpredicting themoisture migration in regular solids during mass-diffusioncontrolled-air dehydration. Equations were presented for moisture at thecenterandfor the massaverage moisture. Theapplicability of these equations was discussed for the spherical geometry employing grain-drying data from publishedliterature. The equations apply only tothe falling rate period ofdehydration in the presence orabsence ofanappreciableconstant rate periods.

INTRODUCTION

Heat transfer equations and charts havebeenwidelyemployed as analogies to predict the moisture distribution in solid materials in diffusion controlled air dehydration (Sherwood 1929, 1931; Newman1931 a,b; Kilpatrick et al. 1955; Jason1958; Fish 1958; Gorling 1958; Charm1978; Heldman and Singh 1981). Theanalogy between the terms in heat transferand mass (moisture) transfer has been welldescribed by Charm (1978).

Moisture removal from solid materialsis characterized by an initial constant rateperiodand a later falling rate period. During the constant rate period there is asteady flow of moisture from the interiorto the surface where it is evaporated. Theoveralldrying rate is dependent on the rateat which moisture is removed at the surface. As drying proceeds, the rate of migration of moisture from the interior to thesurface diminishes and finally becomesthe controlling factor (fallingrate period).The moisture migration during the diffusion-controlled falling rate period followsthe pattern of temperature change underunsteady state heat transfer. Hence, theanalogies apply essentially to the fallingrate period of dehydration. In situations ofshort constant rate periods, the equationsare applied to the entire period (Charm1978). However, corrections are necessary for the nonuniform moisture distribution at the beginning of falling rate period, when the constant rate period issignificantly long (Sherwood 1929, 1931;Newman 1931a,b, 1936).

Themechanisms of moisture migrationduring dehydration of solid food materialshave been investigated by a number of researchers (Van Arsdel et al. 1973) and anumber of modes other than diffusionalmass transfer have been identified. Clayand other non-porous solid such as wood

have been cited as examples (Sherwood1929) which follow diffusional drying.For fish and for starch jells, the diffusioncoefficient is essentially constant until themoisture content (dry basis) has decreasedto 15% (Jason 1958; Fish 1958). The diffusional models, therefore, apply mainlyfor situations involving moderately severedehydration with no significant change inthe product shape or the diffusivity of thematerial during the process. Mathematicsof diffusion-controlled mass-transfer, although not very complicated, often require the use of a computer. While onemight appreciate such a computation forthe case of heat conduction, charts (Gur-ney and Lurie 1923; Newman 1936; Heis-ler 1947) and tables (Newman 1931b)based on these equations might be adequate for the case of dehydration. However, simplified equations that are easilysolvable would be more desirable thantables and charts because they could beeasily employed in programmable calculations.

The objective of thispaperis to developsimplified relationships that aproximatethe original solutions for the moisture distribution in diffusion controlled air-dehydration of regular solids.

THEORETICAL BACKGROUNDThe unaccomplished moisture ratio

(W), Fourier number (Fo) and the recip

rocal of the mass transfer surface resis

tance ration (Af) are the three major parameters in the equations describing theunsteady-state moisture transport duringdehydration. M is analogous to the Biotnumber under heat conduction. The anal

ogy between heat and mass transfer isgiven in Table I.

The relationships for the moisture ratioat the center of an infinite slab (Wop), infinite cylinder (W^) and a sphere (W0s) canbe represented as follows when the initialmoisture distribution is uniform (and nosignificant constant rate period exists)(Carslaw and Jaeger 1959):

2 sinpnWop= S

"-1 pn + sinpncospn

•cos(M^)-exp(-pn2Fo) (1)

where 6n is the nth positive root of

6 tan B=M (2)

Wq, = 2M 2"-1 (A^+7„2Vo(7n)

exp(-7n2Fo) (3)

where 7n is the nth positive root of

yJAy)=MJ0(y) (4)

2M " (8n2 + (M-l)2)sin(8n)w = 2, —

rla "-1 8n2(8n2 + M(A/-l))

• sin(8nr/a) -exp(-8n2Fo) (5)

TABLE I. ANALOGY BETWEEN THE TERMS IN HEAT AND MASS TRANSFER

Heat transfer Mass transfer

Term Notation Term Notation

Fourier number af/a2 Fourier number Dt/a2Thermal diffusivity a Mass diffusivity DTemperature T Moisture w

Heat transfer Mass transfer surfacesurface resistance ratio it/ha resistance ratio DP/b'Kga

Heat transfer coefficient h Mass transfer coefficient Kg

CANADIAN AGRICULTURAL ENGINEERING, VOL. 25, NO. 1, SUMMER 1983143

TABLE II. VALUES OF R AND S FUNCTIONSf

Infinite slab Infinite Cylinder Sphere

M RD sv Rc Sc Rs Ss

0 1.00 0.00 1.00 0.00 1.00 0.00

2.0 1.18 1.16 1.34 2.56 1.48 4.12

50.0 1.27 2.37 1.60 5.56 2.00 9.48

100.0 1.27 2.42 1.60 5.67 2.00 9.67

200.0 1.27 2.44 1.60 5.73 2.00 9.77

00 1.27 2.47 1.60 5.78 2.00 9.87

where 8n is the nth positive root of

8cot8=l-Af (6)

For sufficiently long drying times(Fo>0.2), the terms in the infinite summation series in Eqs. 1, 3 and 5 convergerapidly and in most cases may be accurately approximated by the first term ofthe series (Heisler 1947; Newman 1931a).Further, at the centre of a cylinder, slabor sphere, x = r= 0. Imposing these conditions, Eqs. 1, 3 and 5 were simplifiedfor center moisture ratios in an infiniteslab or plate,

W0p = Rpexv(-SpFo), (7)

an infinite cylinder,

Wqc = /?cexp(-5eFo), (8)

and for a sphere,

W^ = /?sexp(-5sFo). (9)

where Rp, Rc, Rs, Sp, Sc, Ss are characteristic functions of the reciprocal of the masstransfer resistance ratio as given below:

2 sinp,

P, -I- sin^jcos^!

SP = Pi2

2M

sc =

(10)

(11)

(12)

(13)

(14)

(15)

tCalculated from Eqs. 10-15.

transforms of M were use to generate theregression equations. The functions "arc-tan (m.Af)" and "M/(M + n)," where mand n were integers from 1 to 25 or theirreciprocals, were selected primarily because of their similarities to the R and 5functions. Only the top two significanttransforms were use in the developedequations for simplicity.

RESULTS AN DISCUSSION

Equations for the CharacteristicFunctions

The following were the regressionequations developed to predict R and 5values from M values:

Sc = 6.4533M/(M/2)-0.6486 (19)

Rs = 0.7852arctan(M/2)-

0.1145M/(M + 20) + 0.8782 (20)

Ss = 6.0423M/(M + 2) +

2.7846arctan(M/3) - 0.5549 (21)

The arguments for all the trigonometricfunctions in Eqs. 16-21 were in radianmeasure. The coefficients of determination were greater than 0.999 in each ofthese equations. The mean errors in predicting the R and S values, respectively,by using these developed equations were0.13+ 0.078% and 0.17+ 0.097% for aninfinite plate or slab, 0.046 + 0.031% and0.32 + 0.22% for an infinite cylinder, and0.24 + 0.22% and 0.19 + 0.12% for asphere. Figures 1-3 are plots of thesefunctions for an infinite slab, infinite cylinder and a sphere, respectively. Thepoints in each figure represent the originalvalues of the functions while the curves

(M2 + 7i2Vo(7i)

7i2

2M(8t2 + (MR. =

l^sinSi

(8,2 + M(M- 1))8,

METHODOLOGY

The solution of Eqs. 10—15 is difficult,mainly because they involve some characteristic transcendental and Bessel functions. Sherwood (1929, 1931) simplifiedthe relationships by assuming the surfaceresistance to mass transfer to be negligible(M= oo), while Newman (1931a,b) developed tables to facilitate easier computations. The surface resistance to moisturemigrationwill be far from being negligiblewhen the air velocities and relative humidity potentials are low. In the presentinvestigation, M values between 2 and 200were considered in developing the equations. Greater emphasis has been given tolarger M values because moisture is removed from the surface usually at a ratefaster than it is supplied through internalmoisture migration.

Double precision computer techniqueswere used to generate the first roots ofEqs. 2, 4 and 6, and to computetheR andS values for several M values within therangeof interest(TableII). Stepwise multiple regression of R and S on selected

144

Rp = 0.2376arctan(M)-0.01242arctan(M/7) + 0.4219

Sp = 2.6164M/(Af +2)-0.1434

rc = 0.4453arctan(M/2)-

0.1048M/(M + 22) +1.0056

LU

<>

to

QZ<

1 -

10

(16)

(17)

(18)

20 30

M VALUE

Figure1. The characteristic function curves for an infinite slab.

40

CANADIAN AGRICULTURALENGINEERING, VOL. 25, NO. 1, SUMMER 1983

5 -

4 -LU

-J

2c/> 3 -

QZ<

2 -

20 30

M VALUE

Figure 2. The characteristic function curves for an infinite cylinder.

40 50

Figure 3. The characteristic function curves for a sphere.

CANADIAN AGRICULTURAL ENGINEERING, VOL. 25, NO. 1, SUMMER 1983

represent the values calculated from Eqs.16-21. These figures illustrate the excellent agreement between the original andpredicted values. When greater accuracyis needed, the simplified equations reported (Ramaswamy et al. 1982) for unsteady temperature could be employed.

Mass Average MoistureIn the case of dehydration, the mass av

erage moisture content is of greater importance than the center moisture or themoisture at any other locations. Thesehave been well established (Carslaw anJaeger 1959) based on the center temperatures. These can be represented as follows for an infinite plate (Wmp), an infinitecylinder (Wmc) and a sphere (Wms):

wmo = w{Op

sin SJ

C 1/2

2W="mc — Wo,

sc

t s,1/2

• and

"ms ~~ Wr\I f sin 5S1/2 o 1- < -— - cos 5S1/2 >s I Ss1/2 J

(22)

(23)

•(24)

When tables are not available for the Bes-sel function of first order, Eq. 23 can beapproximated with less than 0.5% errorusing

Wmc = W0c(l - 0.1250S, + 5.208 x lO"3^2- 1.085 x lO^Se3 + 1.351 x 10-65c4)-

0<5C<2.40 (25)

Figures 4—6 are plots of the mass average moisture ratios at the center of aninfinite slab, infinitecylinder and a spherecalculated by using the relationships22-24. The points employed Eqs. 10-15,while the lines employed Eqs. 16-21 forcomputation of R and S for different values of M. These figures illustrate excellentagreement between the original and developed equations for M values greaterthan 2. The correlation coefficient in eachcase between the original and predictedvalues was greater than 0.999. The meanerrors in calculating the mass averagemoisture ratio at different M values, 2 to200, and Fourier analog, 0.2 to 10.0, were0.15 ± 0.089% for an infinite plate orslab, 0.43 ± 0.39% for an infinite cylinder and 0.31 ± 0.36% for a sphere. Therelationships reported by Sherwood (1929,1931) and Newman (1931a) can be obtained by substituting the limiting valuesof R and S values for M = oo (Table II) inthe respective equations.

Early Falling Rate PeriodThe equations described so far apply

only to the first term approximation situation, i.e., when the drying time is rela-

145

<or

UJcr

h-CO

o

<a:LU><

toLO<

12 3 4 5

FOURIER NUMBER

Figure 4. Mass average moisture-Fourier number relationships for an infinite slab.

<or

LUor

tO

o

UJ

orLU><

COLO<2

FOURIER NUMBER

Figure 5. Mass averagemoistureratio-Fourier numberrelationships for a sphere.

tively long. For the early falling rate period time drying (Fo<0.2), the predictedvalues of moisture ratio at the center usingEqs. 10-15 or Eqs. 16-24 deviate significantly from those calculated by expanding the series in Eqs. 1, 3 and 5 to severalterms. The error involved increases as theFourier number decreases below 0.2 and

is also a function of the M value. Thecharts in Kreith (1965) or tables developedby Newman (1931b) could be employedto generate information in this region, or

the simplified equations developed forpredicting the temperature ratio duringshort-term heating of thermally conductive solids (Ramaswamy et al. 1982b)could be employed to get information onthe moisture distribution during the earlyfalling rate period.

Application to Wheat DryingTo test the applicability of the devel

oped equations, data were obtained froman earlier study (Bhargava 1970) on air

drying of wheat (var. Park) under a widerange of operating conditions (air temperatures: 48.9, 37.8, 26.7 and 15.6°C; airflow rates: 0.61, 0.41, 0.10 and 0.025 m/sec; air relative humidity, 25-80%). Thedrying data were reported to be essentiallyof falling-rate type such that the equationsdiscussed so far are applicable to the entiredrying period. Example data for three runsat 37.8°C and 50% relative humidity aregiven in Table III. The characteristic radius of the wheat grain was reported to be7.86 x 10"4m. It was reported that for alltest situations the mass average moistureratio, Wm, could be linearized with respectto time (f>3h) on a semi-logarithmic plot.

For a spherical geometry, the equationfor mass average moisture ratio at negligible surface resistance (M = °°) can beobtained from Eq. 24 using appropriatevalues for Rs and Ss from Table II as,

Wm = 0.607exp(-9.87Fo) (26)

Among the three runs given in Table II,condition under no. 221 represented theone with the highest air flow rate, andhence, minimum resistance to mass transfer. The regression equation relating massaverage moisture ratio with time for thissituation was reported as,

Wm = 0.579exp(-8.5xl0"50 (27)

Since the intercept value is less than thatin Eq. 26, the assumption that the surfaceresistance for this test situation is negligible (spherical geometry for wheat grainassumed) is reasonable. By comparing theslopes of Eqs. 26 and 27, the mass diffusivity can be evaluated asD = 8.5xl0"5(a2/9.87) which is equalto 5.32xl0-12m2/sec. Equation 26 canthen be use to predict the drying curve(Fig. 7).

The calculated slope for experimentalcondition under run no. 222 (air flow rateof 0.41m/sec) was -8.17x lO^sec"1,indicating an increased resistance to masstransfer at the surface. Since we know the

mass diffusivity from Run no 221, an Ssvalue can be calculated as

5S = 0.294(a2/D) = 9.50

M can then be found from Eqs. 6 and 15and the corresponding Rs value can beevaluated from Eq. 20. The M and Rsvalues were 52.8 and 2.00, respectively.Equation 24 can now be used to describethe drying curve.

The slope for run no. 223 (air flow rate0.10m/sec) was -7.83 x lO^sec"1, giving an 5S value of 9.10. M and Rs, calculated as before from 5S, were 25.0 and1.99, respectively. Using Eq. 24, thedrying curve can be established.

146 CANADIAN AGRICULTURAL ENGINEERING, VOL. 25, NO. I, SUMMER 1983

ture, mass average (critical) moisture, surface moisture and equilibrium moisture attime zero (start of the falling rate period).

The value of Wm for the different geometries, as described and verified byNewman (1931a), can also be obtainedfrom the R and S values as shown belowfor an infinite slab (Wmp), for an infinitecylinder (Wmc) and sphere (Wms):

2Sp + M-Sp-2M

MSP

2SC + M-SC - 4M

MSC

2SS+MSS-6M

(29)

(30)

(31)

0 1 2

FOURIER NUMBER

Figure 6. Mass average moisture ratio-Fourier number relationships for a sphere.

The predictioncurves as well as experimental data points for the three runs areshown in Fig. 7. The figure indicates reasonable agreement between the predictedand experimental values. Below a dryingtime of 3 h, the curves have beensmoothed out using tabulated values forthe early falling rate period (Newman1931b).

Equations for Falling Rate Period whenConstant Rate Period Exists

In drying of many common materials,a short constant rate period is evident before the onset of the falling rate period.During the constant rate period appreciablemoisture gradientsmaydevelopwithinthe material. The equations discussed sofar are based on initial uniform distribution of moisture and hence cannot be expected to apply to the falling rate period

in the presence of appreciable constantrate periods. Sherwood (1931) and Newman (1931a,b) have discussed the equations relating to these situations, based onthe assumption of parabolic moisture distribution at the beginning of the fallingrate period. Knowing R and S values,these equations can be evaluated as shownbelow:

"•-{2s}"--{^}"-'(28)

where W"m is the mass average moistureratio at any time t\ Wm is the mass averagemoisture calculated as before, assuminguniform moisture distribution and W'm isthe mass average correction factor givenby Eqs. 29-31 for the three geometries,infinite slab, infinite cylinder and sphere;w'o> w' w' and vve are the center mois

These relationships are more complicated and might be useful when the moisture transport mechanism is proven to beonly by diffusion. The major problem under these situations is getting an estimateof w0 an wa at the start of the falling rateperiod. Since parabolic moisture distribution was assumed, knowing either w0 orwa, the other one could be calculated.Sherwood (1931) and Newman (1931b)have given several useful equations relating the two for the different geometries.

CONCLUSIONS

Simple relationships are provided forpredicting moisture diffusion in regularsolids during the falling rate period of air-dehydration. The developed equations forspherical geometry were found to be applicable for predicting moisture migrationduring air-dehydration of wheat grain.The situation involving combined constant and falling rate periods of drying ismore complicated, but can be describedby using the developed equations.

Bi

TABLE III. DRYING DATAt FOR WHEAT GRAIN (ADAPTED FROM BHARGAVA (1970)) C

D

FoMoisture ratio (Wm)

Drying time Run no. 221 Run no. 222 Run no. 223 h(h) (0.61 m/sec)* (0.41 m/sec)* (O.lOm/secW

1.000

Kg

0 1.000 1.000k

1 0.531 0.556 0.586J

2 0.366 0.366 0.385 L

3 0.233 0.260 0.2804 0.168 0.189 0.2095 0.124 0.140 0.1576 0.093 0.106 0.120 M7 0.069 0.080 0.0928 0.050 0.059 0.070 #9 0.036 0.043 0.052

tTemperature andrelative humidity of air were 100°Fand 50%, respectively. StAir flow rate. T

CANADIAN AGRICULTURAL ENGINEERING, VOL. 25, NO. 1, SUMMER 1983

SYMBOLS

= significant dimension (radius of acylinder or sphere, or half-thicknessof slab)

= biot no. (ha/k)= specific heat= mass diffusivity= ourier no. (Dt/a2 or at/a2= surface heat transfer coefficient= mass transfer coefficient= thermal conductivity= essel function of order 0 or 1

= thickness or half-thickness of a slabdepending on it being dehydratedfrom one side or both sides,respectively

= reciprocal of mass transfer resistanceratio (a'KgalD)

= a function of M= radius of a cylinder or sphere= a function of M

= temperature

147

t

w

W'

W" =

a

P7

5

8'

5o"

<or

UJor

H-

o

LUO<orUJ><

in<2

l.U

0.8

0.6\ A

*3k °RUN No. 221

RUN No. 222

v\ D RUN No. 223

0.4

0.2 \R\

0.1

0.08

0.06 -

0J0A

i i i i 1

\0

01 23456789

DRYING TIME(hr)

Figure 7. Moisture loss during drying of wheat grain.

10

= time

= dimensionless moisture ratio =

(w-we)/(wrwe)= mass average correction factor from

Eqs. 29-31mass average moisture ratio

= moisture (dry basis)= moisture (dry basis) at the beginning

of the falling rate period whenconstant rate period exists

= distance from the coldest plane of aslab

= thermal diffusivity (k/°Q= root of Eq. 2= root of Eq. 4= root of Eq. 6= equilibrium constant between solid

phase and vapor phase moisture= density

e = equilibrium moisture at the dryingconditions

i = initial

m = mass average

p = slab or plater = radial distance from the center

s = spherex = distance from the coldest plane of a

slab

0 = center of cylinder or sphere or coldestplane of slab

Subscriptsa = surface

c = cylinder

REFERENCES

BHARGAVA, V. K. 1970. Drying of wheatgrain in thin layers. M.Sc. Thesis, Department of Agriculture Engineering, Univ. ofBritish Columbia. May, 1970.

CHARM, S. E. 1978. The fundamentals offood engineering. 3rd ed. AVI Publ. Co.,Westport, Conn.

CARSLAW, H. S. and J. C. JAEGER. 1959.Conduction of heat in solids. 2n ed. OxfordUniv. Press, London, U.K.

FISH, B. P. 1958. Diffusion and thermodynamics of water in potato starch gel. Fundamental aspects of dehydration of foodstuffs. Macmillan Co., New York.

GORLING, P. 1958. Physical phenomena during the drying of foodstuffs. Fundamentalaspects of dehydration of foodstuffs. Macmillan Co., New York.

GURNEY, H. P. and J. LURIE. 1923. Chartsfor estimating temperature distribution inheating or cooling solid shapes. Ind. Eng.Chem. 15(11): 1170.

HEISLER, M. P. 1947. Temperature charts forinduction and constant temperature heating.Trans. ASME.69:227.

HELDMAN, D. R. and P. SINGH. 1981.Food process engineering. 2n ed. AVI Publ.Co., Westport, Conn.

JASON, A. C. 1958. A study of evaporationand diffusion processes in the drying of fishmuscle. Fundamental aspects of the dehydration of foodstuffs. Macmillan Co., NewYork.

KILPATRICK, P. W., E. LOWE, and W. B.VAN ARSDEL. 1955. Tunnel dehydratorsfor fruits and vegetables. Adv. Food Res.6:360.

KREITH, F. 1965. Principles of heat transfer.International Textbook Co., Scranton, Pa.

NEWMAN, A. B. 1931a. The drying of porous solids: diffusion and surface emission

equations. Trans. AIChE. 27:203.NEWMAN, A. B. 1931b. The drying of po

rous solids: diffusion calculations. Trans.

AIChE., 27:190.NEWMAN, A. B. 1936. Heating and cooling

rectangular solids. Ind. Eng. Chem.28(5):545.

RAMASWAMY, H. S., K. V. LO, andM. A.TUNG. 1982a. Simplified equations to predict unsteady temperatures in regular conductive solids. J. Food Sci.

RAMASWAMY, H. S., K. V. LO, andM. A.TUNG. 1982b. Unsteady temperature inshort-time heated regular thermally conductive solids. Paper presented at the 25th Annual Conf. of CIFST. Montreal, Que.

SHERWOOD, T. K. 1929. Drying of solids.I. Ind. Eng. Chem. 21:12.

SHERWOOD, T. K. 1931. Application of theoretical diffusion equations to drying of solids. Trans. AIChE. 27:190.

VAN ARSDEL, W. B., M. J. COPLEY, andA. I. MORGAN, Jr. 1973. Food dehydration. 2nd ed. Vol. 1 AVI Publishing Co.,Westport, Conn.

148CANADIAN AGRICULTURAL ENGINEERING, VOL. 25, NO. 1, SUMMER 1983